Cylindrical electro-mechanical transducer



R. A. LANGEVIN CYLINDRICAL. ELECTRO-MECHANICAL -TRANSDUCER Filed April 5, 1950 Oct. 16, 1956 2 Sheets-Sheet 1 INVENTOR. ROBERT A. LANGEVIN y Oct. 16, 1956 R. A. LANGEVIN 2,767,387

` CYLINDRICAL ELECTRO-MECHANICAL TRANSDUCER Filed April 5, 1950 2 Sheets-Sheet 2 DIELECTRIC LOSS FACTOR MAXIMUM EFFICIENCY, e, IN PERCENT AssuMING No MECHANICAL I ossEs a B 8 THICKNEss TOYDIAMETER RAI-Io, t/da INVENTOR. FIG. 6 ROBERT ALANGEVIN ATTORNE nited States Patent CYLINDRICAL ELECTRO-MECHANICAL TRANsnUcER Robert A. Langevin, Willoughby, Ohio, assignor, by mesne assignments, to Clevite Corporation, Cleveland, Ohio, a corporation of Ohio u f This invention relates to a transducer for operation in a medium such as water and for transducing therein between electrical and mechanical types of energy. Specifically, the transducer is useful for translating electrlcal energy into mechanical energy or for translating mechanical energy to electrical energy.

Transducers of the type mentioned above are Widely Vused for translating electrical energy to mechanical energy or vice versa. Specifically, piezoelectric materials and magnetostrictive materials have been used for this purpose. More recently barium titanate, when treated in a particular manner, has been found to have transducing properties which are quite beneficial in certain applications.

Also, it has been known for some time that a cylinder has desirable transducing characteristics for certain applications. It is diicult, however, to provide a cylindrical transducer with the piezoelectric crystals of the prior art for the reason that, due to their characteristics of growth, it is impossible to provide a cylindrical transducer which moves radially at all points. Such a i device can be approximated by the use of single crystals if a large mosaic of crystals is'used, but this provides a very expensive structure. The use of magnetostrictive materials for the same purpose also has certain inherent disadvantages. Onl the other hand, it is known that polycrystalline barium titanate, which has been vitrified and treated in a manner to provide transducing properties, can be made toV have a radial action at all points ofthe `surfacewhen in a cylindrical form. The use of .a cylindrical transducer in a liquid such as water, for example, can provide a directional characteristic which is symmetrical for all points around the axis of the cylinder. It, therefore, becomes very desirable in many applications to utilize a tube 'of electro-mechanically sensitive dielectric material of the nature of barium titanate to provide a transducing action in 'a liquid, either for changing electrical energy to mechanical or acoustical energy in Vthe liquid or for changing'such acoustical energy vin the liquid into an electrical form.

Applicant has discovered that, for transducers ofthe type here under consideration, decided-advantages can be obtained when a vhollow cylindrical transducer element "is used if the ratioof the wall thickness (of the cylinder to its diameter isy chosen to be within a particular range or preferably to have substantially Ia given value for the particular material and under the conditions in which it is used. Specically, it has been found that `the maximum efficiency which suchV a `transducer -can have is provided if the ratio `of thickness toroutrside diameter is chosen in a manner'which' will. be set forth in detail hereinafter.

It is an object of the inventionto'provide an improved transducer for use in changing electrical energy to mechanical energy, or vice versa. w c

" It is still another "object of,"the`invention to provide a transducer in the form of a hollow cylinder and which ICC , 2 has substantially the maximum' possible eciency for the transdueing A material utilized.

It is a specific object of the invention to provide aV hollow cylindrical transducer for operation within a hquld with the outside surface of the cylinder acoustically exposed to the liquid and with the inside surface of the cylinder substantially acoustically shielded from the liquid and in which the wall thicknens is so chosen with reference to the diameter of the cylinder as to provide the best eciency in operation.

In accordance with the invention, a transducer for operation in a medium and for transducing therein between electrical and mechanical types of energy comprises a hollow cylinder of electro-mechanically sensitivedielectric material having a radial and tangential activity of opposite sign. ln accordance with a preferred embodiment of the invention, the cylinder comprises a major portion of vitried barium titanate material which has approximately the characteristics of substantially pure barium titanate which has been iired to a temperature within the range of 1300 C. to 1400 C. The trasducer of the invention also includes means, including a provision for causing the outer surface of the cylinder to be exposed to the above-mentioned medium and the inner surface to be acoustically shielded from the medium when the cylinder is immersed in the medium and electrodes on the material, for applying one of the above-mentioned types of energy to the cylinder and for deriving the other of the above-mentioned types of energy therefrom. The cylinder is proportioned to have a ratio of thickness to outside diameter which provides substantially maximum eiciency for the material used and for the conditions under which it is operating.

For a better understanding of the present invention, together with other and further objects thereof, reference is had to the following description taken in connection with the accompanying drawings and itsV scope will be pointed out in the appended claims.

. Fig. 1 of the accompanying drawings illustrates a transducer in accordance with the invention which can be utilitzed either for translating electrical energy to mechanical energy or for translating mechanical energy to electric energy; Fig. 2 represents a section through the transducer 10 of Fig. 1; Figs. 3, 4, and 5 are circuits used in a mathematical description of the proportioning and operating characteristics of the transducer of Fig. 1; while Fig. 6 comprises curves showing certain of the operating characteristics of the transducer of Fig. l. Y

Fig. l of the drawings illustrates a preferred embodiment of the invention. Here a transducer 10 `is illustrated in the form of a hollow cylinder of electro-mechanically -sensitive di-electric material which may be barium titanate. This cylinder has an electrode 1l on the outside surface and an electrode 12 on the inside surface thereof and, when :in operation is adapted to be immersed in a yliquid such as water 13. The arrangement includes means for causing the outer surface ,of the cylinder to be acoustically exposed to the water 13 and for causing the `inner s,urface of thezncylinder to be acoustically shielded' from the water when the cylinder is immersed in the water. Specifically, plugs 15 and 15 are provided for the cylinder and these plugs may be Icomprised of cork, rubber, or any other suitable material. An electrical generator 16 isV shown for applying electrical energy to -the electrodes 11 and 12. Also, a microphone 17 is includedin the liquid 13 in order to generate electrical currents corresponding to the mechanical energy introduced into the liquid by the opera- 4tion of. thecylinder 10 as a transducer for Vconverting the electrical energy of generator 16 to mechanical energy, or specifically to sound energy, in the water 13.

This sound energy, of course, may be of any frequency depending upon `the'frequency of the input signal provided by generator 16 and the resonanty characteristics of the transducer 1t) and may, of course, be very much above any audible frequency. A reproducer 19 is illustrated for reproducing or utilizing the venergy from microphone 17. The reproducer 19'could, for example, be a cathode-ray tube or a loud-speaker or any other reproducing device depending upon the purposefor which the apparatus is to be used.

In considering the operation of the device of Fig. l as so far described, it will be seen that electrical energy, generated by the generator 16, isr applied to the electrodes 11 and 12 thereby causing, underl the conditions to be described in more detail hereinafter, a mechanical action within the cylinder which in turn causes sound waves to be propagated into the water 13. These sound waves are picked up by the microphone 17 and reproduced on the reproducer 19 in any desired manner. As described, therefore, the arrangement of Fig. 1 is one for changing an electrical input to transducer 10 into a mechanical output which comprises the sound energy propagated into the water 13.

Alternatively, the arrangement of Fig. l can be used in the reverse manner. ln'order to 'illustrate this use of the arrangement, switches 22 and 23 are shown. When these switches are in their dotted position, the microphone or transducer 17 is connected to a generator 24 and the electrodes 11 and 12 are connected to a reproducer 25. Under these conditions, the generator 24 is eective to excite the device 17 and produce sound energy in the liquid 13. This sound energy impinging upon the cylinder 10 is transformed by the cylinder 10 into electrical energy and means, including electrodes 11 and 12, are provided for deriving and utilizing this electrical energy. Specifically, the electrical energy is reproduced by reproducer 25 which again may be an oscillograph or any other device which can be actuated by the electrical energy.

In order to explain in detail applicants novel method of proportioning the transducer 10, it is deemed desirable to set forth in a mathematical manner the steps which must be followed and the conditions which must be met. Fig. 2, therefore, illustrates a section through the transducer lit and certain parameters lutilized in the following mathematical analysis are indicated thereon. The electrodes and other portions of the Fig. l arrangement have been omitted for sake of simplicity.

If the transducer of Fig. 1 is operated with the switches 22 and 23 in the positions illustrated in the drawing, electrical energy is supplied to the transducer 10 from the generator 16 and mechanical energy is derived from the transducer 16 through the medium of the water 13. Under these conditions, the electro-mechanical behavior of such a cylindrical tube transducer can be described up through the radial resonance of the tube by an equivalent circuit of the type illustrated in Fig. 3. The symbols used in Fig. 3 have the following significance:

C-Blocked capacity in farads.

Rei-Resistance taking account of dielectric loss. Specifically if D is the dielectric loss factor at a frequency, f,

D=5 where Xc=2dl0 and w=27rf N Transducer ratio in Newton/volt. Cm-Volume compliance in metersl/Newton/meter2 (electrically short circuited). M-Inertance in kilograms/meteri. Rin-Resistance in rn. k. s. acoustical ohms taking account of mechanical losses. f RL-Resistance in m. k. s. acoustical ohms representing effect of water load on the outside-lateralsurfaceof the tube. 'i

In order to calculate the equivalent circuit constants of the subject transducer, lit is convenient to neglect the losses and transforin'the circuit ofv Fig. 3 into 'that of Fig. 4. The circuit of Fig. 4 is the equivalent at all frequencies of that of Fig. 3 (neglecting the losses) provided certain relations are satisfied between the constants 0f the two circuits. These relations are:

Utilizing Equations l, 2, 3 and 4, the constants of the circuit of Fig. 3- are easily calculated providing certain properties of the transducing material used in cylinder 10 are known. `Cylinder 10 is comprised, in a preferred form, substantially solely of barium titanate which has been vitrified by firing to a temperature within the range of l300 C. to l500 C., and which has thereafter been polarized by the application of a high unidirectional potential between the electrodes 11 and 12 in a manner which is generally understood by those skilled in the art. It will be understood that some impurities in addition to barium titanate are frequently included in the material as a practical necessity, but, in general, if the composition is at least barium titanate the material willy generally have the following properties within the percentage variations designated:

meter volts/ Newton.

gaa 12.6 103i20% meter volts/ Newton. Dielectric constant, K 1700i20%.

Poisson ratio, a .31i10%.

All above values taken at 25 centigrade.

Utilizing the above constants, therefore, it is now possible tov obtain an evaluation of the right-hand portions of Equations l to 4.

Specifically, the electrical capacity Ce of a cylindrical tubeV fully lelectroded on its inside and outside surface is given by thel formula for coaxial cylindrical conductors separated by a medium of dielectric constantK. This equation is as follows:

e-loglo kfla/di (5) where di and do are the inside and outside diameters of the tube respectively. If t is the thickness of the tube and notingrthat di/do=1-2t/do, it is clear that the capacity per inch of length can be represented solely as a function of the ratio t/do which is the thickness of tube 10 divided by the outside diameter of the tube.

The following relates to the evaluation of the transducer ratio Na. If the tube of Fig. l is subjected to increments of hydrostatic pressure P0 Newtons/meter2 on the outside there are set up in the material radial and tangential stresses Tr and To which by virtue of the symmetry of the structure can depend only on r and not on 0. Observing that the piezoelectric modulus in the radial direction is everywhere equal to gas and, normal to the radius, to gsi, it follows that the open circuit voltages developed by the stresses Tr and T@ are given respectively by:A

,up farad/inch of length b Waltham .6

Expressions for Tr and Te lin terms of. theradius, '17, of the'tube and P5 are vreadily obtained from elementary elastic theory. (Cf. Timoshenko, Theory of Elasticity, p. 56; McGraw-I-Iill, 1934.) They are It should be observed that because of the opprositehsign'su ofggsa and gsi and because |g33l l2ga1{ for some critical ratio of inside to outside diameter the ratio Na for such cylindrical transducers will be 0.

The following relates to therevaluation of the volume compliance Cv. y l

If eristheradialy strain 'and if u is the displacement of the tube in the radial direction Vas `a result'of the stresses set up by the increment of hydrostatic force,``

and er=SD(T1-o'Te). In additio'n'let v and e@ be the tangential components of displacement and strain respectivslyf Then But E zf l a; l must be identicallyppzero for all and r which can only bef true ii-` rqej=u Since e6=SD(T9-Tr) ,a direct calculation using the previous `values of Tr and ,Ta gives Then .the displacement of the outer radius of the cylinf Having now calculated Ce, Na, ,and Cv (Equations 5, 8, and 12, respectively) as functions of the fundamental constants of the material and the dimensions-of the struc# ture, -the constants C, N, and Cm of Fig. 3 are'known by an application of Equations l, 2, 3, '4L

It ispnow possible to turn to the circuit of Fig, V3 and derive an equation for the eliiciency of theV transducer 10 driving a water load and with its inner surface shielded from the water. For resonant operation, the reactance of Cm and M exactly cancel and by transferringRM and RL to the electrical side of the' transducer there results the equivalent circuit of Fig. 5. f.

,Throughout this discussion it is supposed thatfallV theelementsin the equivalent circuit Fig. are` linear.

NQW, if a'voltage V, be applied to the terminals of the transducer as represented by the equivalent .circuit of..

arenas# 5u' dr==uh so that the volume compliance of fthecylinder is If now we define, as is usual, the eiciency, 0, of the transducer as the ratio of the power radiated to the real power supplied, then .NZRL o: (Rift-R102 N213!I N2RM 1 ,V (KRM-l-Rr'g),2 (RM` +RL)2"\ RD It is now desirable toV re-express this equation for the efficiency, 0, in terms of more familiar parameters. One simple approach to this problem is the following: The dissipation factor, D, of the dielectric is given by the relation r n v where wr is 21r times the resonant frequency of the trans-1 ducer. p

N is, as has been shown,.calculable.in terms of constants of the material and dimensions of the transducer structure.

. If Q0 is the unloaded (Rr.=o) mechanical Q of the transducer represented in Fig. 1, then =,,M M Q., r I If QL is the loaded Q(RL70) of the transducer of Fig. 1

Now referring to the equivalent circuit of the-transducer the coupling coeicient, k, is give'n by the relation N201..` Y lez- N -ZCM C (20).

from which we and that C' 1 -k N2CM" k2 (21) and the efficiency, 6, can finally .be expressed in the form:

The ratio is not` calculablefromtheoretical considerations sincef the'u'nloaded'Q, Qa, necessarily/includes mounting losses,

cement losses, etc. On this tb'asis QL/Qn should properly be considered a hind of gure of merit of a given transducer design. H

Rewriting Equation 22 in the form,

1 sat1. L52 (23) l' QaA Ak2 QL it is clear that if in the right-hand side of this equation wereplace Qr. by, say, QL which is the loaded Q assuming no mechanical losses; it will always be true that QfLQL so that l gives an upper bound to the efficiency of such a transducer in the presence of a dielectric loss factor, D.

In order to obtain numerical values for the maximum eciency guregiven in the last expression it is necessary to know the values of the coupling coeflcient, k, and the loaded Q, QL (assuming no mechanical losses). These can be obtained in thefollowing way.

The coupling coeicient, k, is also expressible in terms of the constants of the circuitV of Fig. 4 as can be shown by a direct application ofthe transformation equations. It' is, in fact, given by the expression EM N C C Now, expressions for all of the constants of the righthand side of this equation have already been obtained (Equations 5, 8, and l2, respectively). Inserting them into this equation for the coupling coeliicient, there results where i inside diameter 2a T dz/do 'outsid'e diameter 2 b Naties, that 1 reaetance of compliance at resonance pwVw 'reload resistance A wrolll where wr is 21r times the radial-resonant frequency and A is the lateral area of the tube (end effects are neglected).

Forpu oses ofcaleulatinglQit isffeasiblejtofn l effect of the elastic coupling to the length Inode on the radial frequencyconstantsothat yit'is`igierrnfis:s ist'il'e to write where B is a constant and dm is the mean diameter.

A suitable value of B can be obtained from the elastic constants of the material by `the relation 4E .l D B \./.12 PQEIS (cf.: Rayleigh, Theory of Sound, Ivols. I and II, 1945). With these assumptions and using the equation already developed for the volume compliance there results which it is clear is a function only of the ratio t/do.

Finally, now, inserting values of the constants of the material into the expressions obtained above for k2 and QL, and using these results in the previously obtained expression for maximum eciency HGr-@et there result curves of the form shown in Fig. 6. Here, for the sake of illustrating the principles, curves have been drawn for dielectric loss factors 'of 1 and l0 percent. Several points are at once evident:

l. Loss factors greater than say l percent seriously impair the eiciency of ceramic cylindrical tube transducers.

QL found in Equation 28 in the last term of the denominator in Equation 30 the following expression is obtained:

i It will be s een that the value of the Expression 30 which represents the maximum efficiency of a cylindrical tube transducer will be a maximum when the` expression of 3,1l is a minimum. Therefore, in accordance with the teachings of the invention, the tube 10 of Fig. l is proportionedy in such a manner as to provide a minimum in the Expression 31. It will be seen by reference tothe curves of Fig. 6 that two dilferent conditions are represented, namely one in which the dielectric losses are 1% and one in which the dielectric losses are 10%. Curves for the conditions in which the dielectric losses lie between 1% and 10% have generally the same form and lie between two curves which are actually shown in lFig. 6. From anl inspection ofthe curves of Fig. 6, it will be seen that the, maximum eiciency is a maximum or is optimumif the ratio of (thickness to diameter is substantially 0.48 and that another maximum is provided if thisl ratio'is substantially 0.125. Also, from an inspection of the curves of Fig. 6,/

this maximum is seen to occur at the same point regardless of. the Value ofthe dielectric loss for the two conditions i shown andithas been determinedfthat the ratio is sbstantially the same for allintermediate values of the fdielecifV tric loss facto'D. lFurthermore, 'it will be seen that theA efficiency of the transducer is maintained safely at a high value if the ratio of` thickness to outer' diameter is 9 the range of 0.03 and 0.2 or if 'it is the range of 0.4 and 0.49. It is furthermore apparent that the'eficiency is zero for the transducer under consideration for all values of dielectric loss factor D if the ratio of thickness to outer diameter is substantially 0.28. From the above, it will be seen that, regardless of the dielectric loss factor of the material, substantially the highest possible operating eiciency will be obtained from Ythe transducer of Fig. 1 if either ratio of thickness to outer diameter is selected to provide optimum efficiency as taught by the above description of the invention.

While there have been described what are at present considered to be the preferred embodiments of the invention, it will be obvious to those skilled in the art that various changes and modifications may be made therein without departing from the invention and it is, therefore, aimed to cover all such modifications and changes as fall within the true spirit and scope of the invention.

What is claimed is:

l. A transducer for operation in a medium and for transducing therein between electrical and mechanical types of energy comprising: a hollow cylinder of electromechanically sensitive dielectric material having a radial and a tangential activity of opposite sign; means, including provisions for causing the outer surface of said cylinder to be acoustically exposed to said medium and the inner surface to be acoustically shielded from said medium when said cylinder is immersed in said medium and electrodes on said material, for applying one of said types of energy to said cylinder and for deriving the other of said types of energy therefrom; said cylinder having a ratio of thickness to outside diameter which substantially provides a minimum of the expression,

wherein:

k denotes the electromechanical coupling coefficient of said material;

wr is equal to 2 times the radial resonant frequency of the transducer;

CM denotes the volume compliance of said cylinder in meters3/Newton/meter2; and

A is the lateral area of said cylinder in meters,

2. A transducer for operation in a medium and for transducing therein between electrical and mechanical types of energy comprising: a hollow cylinder having a ratio of thickness to outer diameter within the range of 0.03 and 0.2 and of material containing a major portion of vitrified barium titanate with a density of substantially 5 .6 X 103 kilograms per cubic meter, an open-circuit compliance of substantially X 10"12 meters squared per Newton, a radial modulus of substantially 12.6 X 10-3 meter volts per Newton, a tangential modulus of substantially -5 .2 X 10-3 meter volts per Newton, a dielectric constant of substantially 1700 and a Poisson ratio of substantially 0.31; and means, including provisions for causing the outer surface of said cylinder to be acoustically exposed to said medium and the inner surface of said cylinder to be acoustically shielded from the surface of said medium when said cylinder is immersed in said material and electrodes on said material, for applying one of said types of energy to said cylinder and for deriving the other of said types of energy therefrom.

3. A transducer for operation in a medium and for transducing therein between electrical and mechanical types of energy comprising: a hollow cylinder having a ratio of thickness to outer diameter of substantially 0.125 and of material containing a major portion of vitried barium titanate with a density of substantially 5.6 X 103 kilograms per cubic meter, an open-circuit compliance of substantially 10 X 10.12 meters squared per Newton, a radial modulus of substantially 12.6 X 10-3 meter volts per Newton, a tangential modulus of substantially 5.2

0.31; and means, including provisions for causing the outer surface of said cylinder to be acoustically exposed to said medium and the inner surface of said cylinder to be acoustically shielded from vthe surface of said medium when said cylinder is immersed in said material and electrodes on said material, for applying one of 4said types of energy to said cylinder and for'deriving the other of said types of energy therefrom.

' 4. Ay transducer. for operation in a medi-um and for transducing therein between electrical and mechanical types of energy comprising: a hollow cylinder having a ratio of thickness to outer diameter within the range of 0.4 to 0.49 and of material containing a major portion of vitried barium titanate with a density of substantially 5.6 X 103 kilograms per cubic meter, an open-circuit compliance of substantially 10 X 10-12 meters squared per Newton, a radial modulus of substantially 12.6 X 10-3 meter volts per Newton, a tangential modulus of substantially -5.2 X 10*3 meter volts per Newton, a dielectric constant of substantially 1700, and a Poisson ratio of substantially 0.31; and means, including provisions for causing the outer surface of said cylinder to be acoustically exposed to said medium and the inner surface of said cylinder to be acoustically shielded from the surface of said medium when said cylinder is immersed in said medium and electrodes on said material, for applying one of said types of energy to said cylinder and for deriving the other of said types of energy therefrom.

5. A transducer for operation in (a medium and for transducing mechanical energy to electrical energy comprising: a hollow cylinder having a ratio of thickness to outer diameter of substantially 0.48 and of material containing a major portion of vitrifed barium titanate with a density of substantially 5.6 X 103 kilograms per cubic meter, an open-circuit compliance of substantially l0 X 1012 meters squared per Newton, a radial modulus of substantially 12.0 X 10-3 meter volts per Newton, a tangential modulus of substantially -5 .2 X 10*3 meter volts per Newton, a dielectric constant of substantially 1700 and a Poisson ratio of substantially 0.31; means, including provisions for causing the outer surface of said cylinder to be acoustically exposed to said medium and the inner surface of said cylinder to be acoustically shielded from the surface of said medium when said cylinder is immersed in said medium, for applying mechanical energy to said cylinder; and means, including electrodes on said material, for deriving electrical energy from said cylinder.

6. A transducer for operation in a medium and for transducing therein between electrical and mechanical types of energy comprising: a hollow cylinder having a ratio of thickness to outer diameter within the range of 0.03 and 0.2 and composed substantially only of barium titanate which has been vitried by firing to a temperature within the range of 1300 C. to 1400" C.; and means, including provisions for causing the outer surface of said cylinder to be acoustically exposed to said medium and the inner surface of said cylinder to be acoustically shielded from the surface of said medium when said cylinder is immersed in said material and electrodes on said material, for applying one of said types of energy to said cylinder and for deriving the other of said types of energy therefrom.

7. A transducer for operation in a medium and for transducing therein between electrical and mechanical types of energy comprising: a hollow cylinder having a ratio of thickness to outer diameter within the range of 0.4 and 0.49 and of material composed substantially solely of barium titanate which has been vitriied by being tired to a temperature within the range of 1300 C. to 1400 C.; and means, including provisions for causing the outer surface of said cylinder to be acoustically exposed to said medium and the inner surface of said cylinder to be acoustically shielded from the surface of said medium 2,076,330 Wood Apr. 6, 1937 2,402,515 Wainer June 18, 1946 2,420,864 Chilowsky May 20, 19,47 2,486,560

Gray Nov. 1, 1949 OTHER REFERENCES Article by Roberts, Physical Review, June 15, 1947, pages 89.0,895.

Article by Mason, Bell Laboratories Record, August, 1949, pages 285-289. 

